It is by virtue of several precisely matching characteristics - a uniquely contoured scratch line and a small blemish formed by excess dried paint on the front face of the pistol body, as well as the relative positions of the grooved regions on the silver nozzle ring - that the Type 2 phaser pistol offered in Hollywood Auction 56 can be uniquely identified as the specific prop seen in a screenshot from the highly memorable time-travel episode "Assignment: Earth", the last episode of the 2nd season of TOS.
In addition to this main finding, the Profiles write-up specifically discusses the use of screenshots from other TOS episodes to match particular features/ contours / design elements observed to exist on the prop for sale; but is very clear in asserting that it is the design characteristic alone - such as the hand grip geometry or a mold flaw in the rear fin design, etc. - that has been matched to a screenshot from a different episode; and not the unique TOS phaser prop itself. In particular, the auction description notes that a tiny mold flaw/ridge line deviation in the rear fin appearance was matched on-screen in "Spock's Brain"; the integrated "one piece" handle and main body construction of the pistol component of this prop was matched on-screen in "The Cloud Minders"; and the design of the hand grip section with an elongated central groove extending along the entire length of the handle was matched on-screen in "Plato's Stepchildren".
The "mold flaw" detail match observed on the "Spock's Brain" screenshot is a particularly exciting discovery. By comparing the two props seen in the annotated image presented immediately below, it is apparent that they are not the same unique phaser pistol since, along with other details, the silver nozzle was attached to the front of the pistol body differently on the two pieces , and thus the position of the grooved regions on the nozzle ring relative to the rest of the body details do not match. Yet the two fiberglass pistol bodies clearly came from the same mold and feature the same ridge line deviation/flaw in the rear fin area that could only be introduced into the appearance of both props from the very same mold! The crooked ridge lines share the precisely identical geometry, and thus one can conclude that there exists a precise match between these two phasers on other appearance characteristics such as contour lines and curvatures which are also imparted to individual props from the mold that was used to fabricate them.
Presented below is the complete Profiles auction catalog description for this extremely rare Original Series hand prop, as well as some additional high resolution imagery. While it does bear the to-be-anticipated paint wear, surface scratches and cracks/seam separation in its fiberglass pistol body; and it is missing the semi-circular acrylic dial and metal faux thumbwheel control that were once attached to the top of its hand phaser; this Type 2 phaser pistol remains an impressively sturdy and very highly presentable prop. The TOS phaser pistol is perhaps the most desirable and iconic hand prop from The Original Series, instantly recognizable to millions of people worldwide, and, it is my opinion that this specimen - one of only two such examples known to have survived from the 60s - could easily serve as the centerpiece in the finest of Star Trek collections.
With the recent impressive auction results realized for TOS memorabilia - an Original Series Type 1 hand phaser selling for $79,950 in December, 2011; the early first season Captain Kirk tunic witnessing a final price of $98,400 in July, 2012; and the TOS phaser rifle from the 2nd pilot reaching a total of $231,000 just a few months ago in April, 2013 - it will be exciting to see how this exceptionally rare screen matched Type 2 phaser pistol performs.